Bipolar disorder (BD) is among the most chronic and severe types of mental illnesses. The disease is a complex neuropsychiatric condition characterized by infrequent but extreme episodes of low (depressed) and elevated (manic) moods. When compared to other mental illnesses, the global health burden of BD is massive: 1-4% of all adults live with this condition, equating to over 17.5 million years lived with the disability (YLDs). Data also indicates that among all mental illness patients, those with BD have the highest rate of suicide.

Treatment for BD has focused principally on the use of lithium, which has been shown to reduce hyperactivity, efficacy of neuron action potential firings, and rescue mitochondria dysfunction. Nevertheless, a more thorough examination of the dynamics of BD is necessary if researchers hope to gain a better understanding of the nuances of this disease.

A rich array of mathematical approaches model the spread of communicable diseases such as influenza, malaria, and dengue fever. However, there is little comprehension of the dynamics of mental illnesses such as bipolar disorder, or whether understanding dynamics might help develop targets for treatment. Given the present interest in dynamics, clearly a role for a deeper mathematical comprehension of the dynamics of BD exists. Our group at the University of Oxford has been using applied maths to develop this idea.

Teaming up with psychiatrists and clinical psychologists, our aim has been to link the observed dynamics of BD, based on noisy patient mood profile data, with mathematics from statistics and dynamical systems theory—colloquially known as ‘mood maths’—to achieve clinically-relevant predictions and a more comprehensive understanding of the disease.

Patients are able to self-report their moods through well-proven and standardized psychological scoring systems. These types of scales include the Becks Depression Inventory, the Altman Self-Rating Mania Scale, and the Quick-Inventory for Depressive Symptomology (QIDS); each measures different aspects of a patient’s risk of BD. Using data from the QIDS scale, which patients regularly report via smartphone or internet-based technologies, we developed autoregressive time series approaches as novel descriptors of patient mood.

Autoregressive time series models use mood scores at previous points in time as predictors to explain current mood. Mathematically, these time series models necessitate the construction of likelihoods, statistical descriptors relating the hypothesis that model \(\textrm{M}\) generated our observed data \(\textrm{D}\). Developing these likelihoods required approaches to manage missing values and errors that are not normally distributed. Our likelihoods take the following form: \[ L(\mathbf D | \mathbf M)=\frac{Y_{1}^{r-1}\left(\frac{r}{\mu_1}\right)^r exp\left(-\left(\frac{r}{\mu_1}\right)Y_{1}\right)}{\Gamma(r)} \] \[ \prod^N_{i=2}\frac{Y_{i}^{r-1}\left(\frac{r}{\mu_i}\right)^r exp\left(-\left(\frac{r}{\mu_i}\right)Y_{i}\right)}{\Gamma(r)} \] where \(r\) is a parameter associated with the underlying probability distribution of the errors between model predictions \((\mu)\) and mood observations \((\textrm{Y})\). \(\Gamma(r)\) is the gamma function \((\Gamma(r)=(r-1)!)\). Time series likelihoods are based on conditional probabilities: the value now \((V_t)\) is dependent on a value at some previous time point \((V_{t-\tau})\). This type of dependency introduces time lag correlations and also requires the development of ways to handle the first observation in a time series. Historically these observations have been excluded from likelihood calculations, but here we condition this observation on the mean of the model predictions.

Based on clinical assessments, patients were grouped into high and low risk (of extreme mood episodes). Across these groups, our time series models highlight different time lag correlation structures in the patients’ time series. More correlation lags above and below a threshold were needed to predict and describe the observed dynamics in the ‘high-risk’ group [2].

But this is only a descriptive approach to BD dynamics, and we really want a more mechanistic link between neurophysiological processes and mood dynamics. Thus, we recently used a ‘relaxation oscillator’ approach to open up this problem (see Figure 1) [1]. As a set of ODEs, relaxation oscillators take the form: \[ \frac{dx}{dt}=y(t)-f(x) \] \[ \frac{dy}{dt}=\frac{-x-a}{b} \] where \(a\) and \(b\) are parameters and \(f(x)=-x-(x^3/3)\). This system of equations is based on a van der Pol oscillator, a well-known descriptor for stable oscillating dynamics in electrical engineering. A relaxation oscillator is perfect for linking BD and underlying fluctuating processes (like neuron firing patterns). It has well-known properties, including the characterization of dynamics by periods of low and high states with rapid ‘relaxation’ between these states. The dynamics can also be stable under certain conditions.

Figure 1. Illustration of predicted relaxation oscillator dynamics for bipolar disorder dynamics. (a) Time series of observed mood scores (based on QIDS scores), (b) predicted independent relaxation oscillator dynamics based on parameters derived from model fit to time series, (c) predicted dynamics based on total derivative (including baseline mood and contribution of oscillator). Adapted from [1].

A single relaxation oscillator might be thought to describe the up and down dynamics of low and elevated moods associated with BD. However, our approach has been more nuanced than this; we ask how oscillators (which capture high or low moods) ‘relax’ from a state of high (or low) mood to a state of average or baseline mood and best capture BD dynamics.

Another aspect of dynamics that is important when modeling BD is the possibility of so-called ‘noise-induced instabilities’. Essentially, these are oscillations that occur in a relaxation oscillator and are driven by noise, rather than deterministic changes. This distinction is important because mood patterns in BD patients might simply fluctuate around a baseline (the ‘steady-state’), making the relaxation oscillator dynamical repertoire ideally suited for investigating BD dynamics.

We use a total derivative of mood changes through time when linking these relaxation oscillators to mood dynamics. This derivative partitions baseline mood and contributions to mood dynamics from oscillators: \[\frac{dM}{dt}=\alpha+\beta\frac{d\mathbf{X}}{dt}\] where \(\alpha\) is baseline mood and \(\beta\) is a scalar relating the relaxation oscillator \((d\mathbf{X} / dt) \) to mood. Using time series methods, we scrutinize the credibility of this model to predict mood dynamics. Specifically, we test many different models where oscillators are coupled, independent, noisy (or not) and conclude that independent oscillators best predict BD mood dynamics across different patients. The departures from the fit of the model to the mood observations are also quite informative. Equally important to our understanding of the dynamics are the levels of baseline mood \((\alpha)\) and endogenous noise, and their contributions to changes in mood dynamics. Endogenous noise is identified as stochasticity associated with parameters in our mood model, and the uncertainty between model fit and the data; each individual patient has patterns of this variability.

Categorizing QIDS-based mood scores into three groups (low, medium, and high) for each individual patient allows us to obtain probability transitions (from one day to the next) with simple Markov chains. Using these stochastic process models, we can then solve for the long-term probabilities (for each of the three groupings) before and after treatment.

This simple metric based on the probability of certain ‘mood states’ (low, medium, or high) has potential for clinical application, particularly if it helps patients understand their mood trajectories more clearly. With greater advances in molecular biology, those involved in 21st century bioscience aspire towards the development of individualized treatments and medicines. Mathematics has a crucial role to play in this; perhaps developing more mathematical approaches to better understand the dynamics of diseases such as BD can set us on a pathway towards this goal.

Michael Bonsall is a professor of mathematical biology and fellow of St Peter’s College at the University of Oxford. He also leads the Mathematical Ecology Research Group in the Department of Zoology. Further details on research in the group are available here.

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